Existence for some vectorial elliptic problems with measure data **)

[1] EXISTENCE FOR SOME VECTORIAL ELLIPTIC PROBLEMS WITH MEASURE DATA 33

F

L

P

V

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(*)

Existence for some vectorial elliptic problems

with measure data (**)

1 - Introduction

The study of elliptic boundary value problems with measure data div (a(x;u(x);Du(x)))ˆm inV

uˆ0 on@V

(

…1:1†

received great attention starting from[2]. In such a paper the scalar case

u:VRn !Rhas been considered and the model operator is the p-laplacian

ai(x;u;j)ˆ jjjp 2ji. Some years later [3], the anisotropic pi-laplacian has been dealt with:ai(x;u;j)ˆ jjijpi 2ji, that is, each component of the gradientDiumay have a different exponent pi; this seems useful when dealing with some re-inforced materials, [11]; see also [10], example 1.7.1, page 169. Let us point out that both [2] and [3] deal with the scalar case u:V!R; as far as the vectorial caseu:V!RNis concerned, some contributions appeared in [7] and [4], where the vectorialp-laplacian is considered. In this paper we deal with the anisotropic

(*) F. Leonetti: Dipartimento di Matematica Pura ed Applicata, UniversitaÁ di L'Aquila, 67100 L'Aquila, Italy; e-mail: [email protected]; P.V. Petricca: Via Sant'Amasio 18, 03039 Sora, Italy.

(**) Received November 17th_{2005 and in revised formMarch 2}nd_{2006. AMS} classifica-tion 35J60, 35D05, 35J55.

vector valued case: Pn iˆ1Di(jDiuj pi 2_D iu)ˆm inV uˆ0 on@V; 8 < : …1:2†

where u:V Rn_!RN _{with n3, N1, V is a bounded open set and}

mˆ(m1_{;. . .;m}N_{) is a finite Radon measure onR}n_{with values intoR}N_{. In this paper} we prove existence of weak solutions for the Dirichlet problem(1.2): in Section 2 we give the precise result, whose proof appears in Section 3. We thank the referee for useful remarks and suggestions.

2 - Notations and statements

We study existence of weak solutions for the Dirichlet problem

Pn iˆ1Di(jDiuj pi 2_D iu)ˆm inV uˆ0 on@V; 8 < : …2:1†

where u:V Rn_!RN _{with n3, N1, V is a bounded open set and}

mˆ(m1_{;. . .;m}N_{) is a finite Radon measure onR}n_{with values intoR}N_{. We need some} restrictions on the exponentspi2; we introduce the harmonic mean

pˆ _n1Xn iˆ1

1

pi

! ₁

and we assume that

p<nand p_(p(n ₁₎1)_n<pi<p(_{n p}n 1) 8i2 f1;. . .;ng: …2:2†

The right hand side of (2.2) can be written as

pi 1 n_p((_np 1)₁₎

ˆpi_pn p_{(n 1)}<1 thus

1pi 1<n_p(₍p_n 1)₁₎pi so we can take exponentsq1;. . .;qnsuch that

and we set qˆmin i qi; qˆ 1 n Xn iˆ1 1 qi ! ₁ ; _q1ˆ1_{q n}1:

In this paper we will prove the following

Theorem2.1. Under the previous assumptions there exists u2W₀1;q(V;RN₎ \Lq_(V;RN_{)with D} iu2Lqi(V;RN) 8i2 f1;. . .;ngsuch that … V Xn iˆ1 XN bˆ1 jDiujpi 2DiubDiWbdLnˆ XN bˆ1 … V Wb_dmb _8W2C1 c (V;RN): …2:3†

In this theoremwe may takenˆ3,p1ˆp2ˆ2 andp32(2;‡1); this example shows that our theoremcannot be derived from[5], [13].

The previous theoremimproves on [9]: we remove the restriction on the support of the measuremand we allow a wider range forpi.

3 - Proof of the Theorem

LetfQkgk2Nbe a decreasing sequence of positive numbers converging to zero. We mollify our measuremand we obtain smooth functions

fkˆmrQk* _m with sup k kfkkL1(V) jmj(R n_)<‡1: …3:1†

We setpˆminipiand we use direct methods in the calculus of variations in order to find uk2 V ˆ n u2W₀1;p(V;RN): Diu2Lpi(V;RN) 8i2 f1;. . .;ng o ; such that … V Xn iˆ1 XN bˆ1 jDiukjpi 2Diub_kDiWbdLnˆ XN bˆ1 … V Wb_fb kdLn 8W2 V: …3:2†

on level sets [2]; we set M(t)ˆ 0 0tL t L L<t<L‡1 1 tL‡1 M( t) t<0 8 > > > > < > > > > :

and we use the following test functionWin (3.2): we fixa2 f1;. . .;Ngand we take

Wˆ(W1_{;. . .;W}N_{) withW}b_{ˆ0 forb6ˆaandW}a _ˆM(ua

k), whereuakis thea-th compo-nent ofukˆ(u1k;. . .;uNk). We define

BL;kˆx2V:Luak(x)<L‡1 ; and we insert the previousWinto (3.2): we get

… BL;k Xn iˆ1 _Diukpi 2_D iuak2dxˆ … V fa kM(uak)dx: …3:3†

SincejM(t)j 1, we can use (3.1) so that the right hand side of (3.3) is bounded by jmj(Rn). Let us look at the left hand side: sincepi2, we have

_Diua kpi 2Diukpi 2 thus we get … BL;k Xn iˆ1 _Diua kpidxc1 8k;L2N: …3:4† Since qi<n_p((_np 1)₁₎pi; we can select u2 0;n(p 1) p(n 1) (0;1)

such thatqiupifor everyi. Now, without loss of generality, we can assume that

qiˆupi for everyiand

Let us remark that 1<qi<pifor everyi2 f1;. . .;ngandq<p<n. Now we follow [3]: we set

lˆ(1 _uu)q

so thatl>1; we use Holder inequality:

… V _Diua kqidxˆ … V _Diua kqi(1‡ juakj) lqi=pi(1‡ juakj)lqi=pidx … V _Diua kpi(1‡ juakj) ldx 2 4 3 5 qi=pi … V (1‡ jua kj)lqi=(pi qi)dx 2 4 3 5 1 qi=pi ˆ X‡1 Lˆ0 … BL;k _Diua kpi(1‡ juakj) ldx 2 6 4 3 7 5 qi=pi  V (1‡ jua kj)lqi=(pi qi)dx 2 4 3 5 1 qi=pi X‡1 Lˆ0 (1‡L) l … BL;k _Diua kpidx 2 6 4 3 7 5 qi=pi … V (1‡ jua kj)q dx 2 4 3 5 1 qi=pi : …3:5†

We keep in mind (3.4) andl>1: we get

… V _Diua kqidxc2 … V (1‡ jua kj)q dx 2 4 3 5 1 qi=pi : …3:6†

We use the anisotropic embedding theorem [12] and we get

_ua kLq_(V)c3 Yn iˆ1 _Diua k1L=qin(V)c4 Yn iˆ1 … V (1‡ jua kj)q dx 2 4 3 5 [(1=qi) (1=pi)]=n ˆc4 … V (1‡ jua kj)q dx 2 4 3 5 (1 u)=(up) : …3:7† We recall thatp<n, so (1 u)=(up)<1=q thus (3.7) gives _ua kLq_(V)c5 8k2N; 8aˆ1;. . .;N: …3:8†

We insert this bound into (3.6) and we get

_Diua

kLqi(V) c6 8k2N; 8iˆ1;. . .n; 8aˆ1;. . .;N: …3:9†

Because of weak compactness, there exists u2W₀1;q(V;RN)\Lq_(V;RN_{), with}

Diu2Lqi(V;RN) 8i2 f1;. . .;ng, si2L

qi

pi 1_(V;RN_{) and h}

i2Lqi(V;R) for every

i2 f1;. . .;ngsuch that, up to a subsequence,

uk*u in W₀1;q(V;RN); …3:10† uk !u a:e:inV; …3:11† Diuk*Diu inLqi(V;RN) 8i2 f1;. . .;ng; …3:12† jDiukjpi 2Diuk*si inL qi pi 1(V;RN) 8i2 f1;. . .;ng; …3:13† jDiuk Diuj*hi0 inLqi(V;R) 8i2 f1;. . .;ng: …3:14† We claimthat hiˆ0 inV 8i2 f1;. . .;ng …3:15† thus Diuk!Diu inL1(V;RN) 8i2 f1;. . .;ng: …3:16†

We prove claim(3.15) following [4]. We fix a smoothv:Rn!RN; later, vwill be chosen as a linear Taylor polynomial ofu. We take cut off functionsf:Rn!Rand

h:RN_{!Rsuch that}

f2C1

c (V;R);

h2C1

c (RN;R);

with 0f1 and 0h1. We takec:RN_!RN_{as follows}

c(z)ˆg(jzj)_jz_zj

where g2C1_{((0;‡1);R) with g and g}0 _{non-negative and bounded; moreover, we} require thatcjsupp (h)ˆ Id . Note that

@ca @zs (z)ˆg0(jzj) zs_za jzj2 ‡ g(jzj) jzj dsa zs_za jzj2 ! thus XN aˆ1 jaXN sˆ1 @ca @zs(z)js0 8z;j2RN: …3:17†

For everyj2 f1;. . .;ngit results that 0 … V jDjuk Djujh(uk v)fdx … V jDjuk Djvjh(uk v)fdx‡ … V jDjv Djujh(uk v)fdx … V jDjuk Djvjpjh(uk v)fdx 0 @ 1 A 1=pj … V h(uk v)fdx 0 @ 1 A 1 1=pj ‡ … V jDjv Djujh(uk v)fdx: …3:18†

Let~pˆmaxipi. Then, using the monotonicity ofz! jzjp 2zand the equalitycˆId on supp(h), we get 0T_kj :ˆ … V jDjuk Djvjpjh(uk v)fdx Xn iˆ1 … V jDiuk Divjpih(uk v)fdx 8~p 1Xn iˆ1 … V hjDiukjpi 2Diuk jDivjpi 2Div;Diuk Divih(uk v)fdx ˆ8~p 1Xn iˆ1 … V hjDiukjpi 2Diuk;Di(c(uk v))ih(uk v)fdx 8~p 1Xn iˆ1 … V hjDivjpi 2Div;Diuk Divih(uk v)fdx ˆ8~p 1 Xn iˆ1 … V hjDiukjpi 2Diuk;Di(c(uk v))ifdx 8 < : Xn iˆ1 … V hjDivjpi 2Div;Diuk Divih(uk v)fdx Xn iˆ1 … V hjDiukjpi 2Diuk;Di(c(uk v))i[1 h(uk v)]fdx 9 = ;: …3:19†

We recall (3.2) and we get Xn iˆ1 … V hjDiukjpi 2Diuk;Di(c(uk v))ifdx ˆXn iˆ1 … V hjDiukjpi 2Diuk;Di(c(uk v)f)idx Xn iˆ1 … V hjDiukjpi 2Diuk;c(uk v)Difidx ˆ … V hmr_Qk;c(uk v)fidx Xn iˆ1 … V hjDiukjpi 2Diuk;c(uk v)Difidx: …3:20† Moreover, (3.17) implies Xn iˆ1 … V hjDiukjpi 2Diuk;Di(c(uk v))i[1 h(uk v)]fdx Xn iˆ1 … V XN aˆ1 jDiukjpi 2Diuak XN sˆ1 @ca @zs(uk v)Divs[1 h(uk v)]fdx: …3:21†

We insert (3.20) and (3.21) into (3.19): we get 0Tj_k:ˆ … V jDjuk Djvjpjh(uk v)fdx 8~p 1 … V hmr_Qk;c(uk v)fidx 8 < : Xn iˆ1 … V hjDiukjpi 2Diuk;c(uk v)Difidx ‡Xn iˆ1 … V XN aˆ1 jDiukjpi 2Diuak XN sˆ1 @ca @zs(uk v)Divs[1 h(uk v)]fdx Xn iˆ1 … V hjDivjpi 2Div;Diuk Divih(uk v)fdx 9 = ;: …3:22†

continuous and bounded: for everyjˆ1;. . .;nwe get 0Tj 1:ˆlimsup k!‡1 T j k 8~p 1 _sup z2RNjc(z)j … V fdjmj Xn iˆ1 … V hsi;c(u v)Difidx 8 < : ‡Xn iˆ1 … V XN aˆ1 sa i XN sˆ1 @ca @zs(u v)Divs[1 h(u v)]fdx Xn iˆ1 … V hjDivjpi 2Div;Diu Divih(u v)fdx 9 = ;; …3:23† 0 lim k!‡1 … V jDjuk Djujh(uk v)fdxˆ … V hjh(u v)fdx; …3:24† lim k!‡1 … V jDjv Djujh(uk v)fdxˆ … V jDjv Djujh(u v)fdx; …3:25† lim k!‡1 … V h(uk v)fdxˆ … V h(u v)fdx: …3:26†

Take the limsup in (3.18). We get, for everyjˆ1;. . .;n, 0 … V hjh(u v)fdx Tj 1 1=pj … V h(u v)fdx 0 @ 1 A 1 1=pj ‡ … V jDjv Djujh(u v)fdx …3:27† whereTj

1 has been estimated in (3.23). For a. e.a2Vand everyjˆ1;. . .;n, we have (see [6] and [1])

lim R!0‡ … B(a;R) ju(x) [u(a)‡Du(a)(x a)]j R dxˆ0; …3:28† lim R!0‡ … B(a;R) jDu(x) Du(a)jdxˆ0; …3:29† lim R!0‡ … B(a;R) jsj(x) sj(a)jdxˆ0; …3:30†

lim R!0‡ … B(a;R) jhj(x) hj(a)jdxˆ0; …3:31† limsup R!0‡ jmj(B(a;R)) Ln_(B(a;R))<‡1 …3:32†

whereLn _{is the Lebesgue measure inR}n_and

… A f(x)dxˆ_Ln1_(A) … A f(x)dx:

We choosevto be the linear Taylor polynomial ofuina:

v(x)ˆu(a)‡Du(a)(x a): Since we are still free to selectf,handc, let

f_R(x)ˆ_R1_nf~x a_R ; ~f2C1 c (B(0;1);R); 0~f1; … Rn ~ f(x)dxˆ1 hR(y)ˆ~h _Ry ; ~h2C1c (B(0;1);R); 0h~1; ~hjB(0;1 2)ˆ1; c_R(y)ˆR~c _Ry ;

withc~ as in the above discussion. Then (3.27) can be written as follows: 0 … B(a;R) hjhR(u v)fRdx Tj_1;R1=pj … B(a;R) h_R(u v)f_Rdx 0 B @ 1 C A 1 1=pj ‡ … B(a;R) jDjv DjujhR(u v)fRdx …3:33† where 0Tj_1;R:ˆlimsup k!‡1 … B(a;R) jDjuk DjvjpjhR(u v)fRdx 8~p 1 ( sup z2RNjcR(z)j … B(a;R) f_Rdjmj Xn iˆ1 … B(a;R) hsi;cR(u v)DifRidx ‡Xn iˆ1 … B(a;R) XN aˆ1 sa i XN sˆ1 @ca R @zs (u v)Divs[1 hR(u v)]fRdx Xn iˆ1 … B(a;R) hjDivjpi 2Div;Diu DivihR(u v)fRdx ) : …3:34†

Recalling (3.28), ... , (3.32) and the definitions ofh_R,f_R, we get lim R!0‡ … B(a;R) hjhR(u v)fRdxˆhj(a); …3:35† lim R!0‡T j 1;Rˆ0; …3:36† 0 … B(a;R) hR(u v)fRdx1; …3:37† lim R!0‡ … B(a;R) jDjv DjujhR(u v)fRdxˆ0: …3:38†

As far as (3.36) is concerned, we give some details in the Appendix. Let us take

R!0‡_{in (3.33): we get, for everyjˆ1;. . .;n,}

hj(a)ˆ0 for a.e:a2V …3:39†

thus our claim(3.15) is proved. Now we have (3.16), thus, up to a subsequence,

Duk!Dua:e:in V: …3:40†

The previous pointwise convergence and the estimate

_jDiukjpi 2_D iuk Lpiqi1_(V)ˆDiuk pi 1 Lqi(V)c7; 8k2N; 8i2 f1;. . .;ng allow us to get jDiukjpi 2Diuk*jDiujpi 2Diu in L qi pi 1(V;RN) 8i2 f1;. . .;ng: …3:41†

Thus we can pass to the limit, ask! ‡ 1, in (3.2) and it turns out thatusolves (2.3).

This ends the proof. p

4 - Appendix

For the convenience of the reader, we give some details about the proof of (3.36). We look at inequality (3.34): it suffices to prove that all the four integrals go to zero whenR!0‡_{. We keep in mind that}

jc_R(z)j c8R …4:1†

0f_R(x)_R1_n …4:2†

then 0 sup z2RNjcR(z)j … B(a;R) f_Rdjmj c8Rjmj(B_R(a_n;R))

and the right hand side goes to zero because of (3.32). Now we deal with the second integral in the right hand side of (3.34):

… B(a;R) hsi(x);cR(u(x) v(x))DifR(x)idx … B(a;R) hsi(a);c_R(u(x) v(x))DifR(x)idx ‡ … B(a;R) hsi(x) si(a);cR(u(x) v(x))DifR(x)idx ˆIR‡IIR: Note that DifR(x)ˆ_R1n Di~f x a R h i₁ R: We recall that ~ c(z)ˆzforz2B 0;1₂ supp (~h) then cR(u v)ˆu v on ju v_R j1₂

thus it is convenient to let

ARˆ x2B(a;R):ju(x)_Rv(x)j>1₂

:

We definevnto be thendimensional Lebesgue measure of the unit ball inRn; thenLn_(B(a;R))ˆv nRnand 0Ln_R(A_nR)2vn … B(a;R) ju(x) v(x)j R dx:

We use (3.28) and we get lim R!0‡ Ln(AR) Rn ˆ0: …4:3†

Now we are able to deal withIRas follows:

0IRˆ _R1_n … AR hsi(a);cR(u(x) v(x)) Di~f x a_R h i₁ Ridx ‡ 1 Rn … B(a;R)nAR hsi(a);cR(u(x) v(x)) Dif~ x a_R h i₁ Ridx …4:4† Ln_R(A_nR)jsi(a)jc8Dif~_L1_(Rn₎‡vnjs_i(a)jD_i~f_L1_(Rn₎ … B(a;R) ju(x) v(x)j R dx:

Using (4.3) and (3.28) we get

lim

R!0‡IRˆ0: …4:5†

As far asIIRis concerned, we argue as follows: 0IIRc8vnDif~_L1_(Rn₎

B(a;R)

jsi(x) si(a)jdx

thus we can use (3.30) and we get lim

R!0‡IIRˆ0: …4:6†

Limits (4.5) and (4.6) guarantee that the second integral in (3.34) goes to zero when

R!0‡_{. In a similar manner we can show that the third integral goes to zero. The} fourth integral is easier to be dealt with. Thus we have established (3.36).

References

[1] L. AMBROSIO, N. FUSCOand D. PALLARA,Functions of bounded variation and free discontinuity problems, Mathematical Monographs, Oxford University Press, New York 2000.

[2] L. BOCCARDO and T. GALLOUET, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal.87(1989), 149-169.

[3] L. BOCCARDO, T. GALLOUETand P. MARCELLINI,Anisotropic equations in L1,

Differential Integral Equations9(1996), 209-212.

[4] G. DOLZMANN, N. HUNGERBUHLERand S. MULLER,The p-harmonic system with measure-valued right hand side, Ann. Inst. H. Poincare Anal. Non LineÂaire

14(1997), 353-364.

[5] G. DOLZMANN, N. HUNGERBUHLERand S. MULLER,Non-linear elliptic systems with measure-valued right hand side, Math. Z.,226(1997), 545-574. [6] L. C. EVANS and R.F. GARIEPY, Measure theory and fine properties of

functions, CRC Press, Boca Raton, FL 1992.

[7] M. FUCHSand J. REULING,Non-linear elliptic systems involving measure data,

Rend. Mat. Appl. (7)15(1995), 311-319.

[8] F. LEONETTI, Maximum principle for vector-valued minimizers of some in-tegral functionals, Boll. Un. Mat. Ital.5-A (1991), 51-56.

[9] F. LEONETTI and P. V. PETRICCA, Anisotropic elliptic systems with measure data, Preprint6, Dip. Mat. Pura Appl. Univ. L'Aquila (2004).

[10] J. L. LIONS,Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier - Villars, Paris 1969.

[11] Q. TANG,Regularity of minimizers of non-isotropic integrals of the calculus of variations, Ann. Mat. Pura Appl. (4)164(1993), 77-87.

[12] M. TROISI, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche

Mat.18(1969), 3-24.

[13] S. ZHOU,A note on nonlinear elliptic systems involving measures, Electron. J.

Differential Equations,2000(2000), 1-6.

Abstract

We study the Dirichlet problem

Pn iˆ1Di(jDiuj pi 2_D iu)ˆm inV uˆ0 on@V; 8 < :

whereu:VRn_!RN_{withn3,N1,Vis a bounded open set andmˆ(m}1_{;. . .;m}N_{)is a}

finite Radon measure onRnwith values intoRN. Under some restrictions on the exponents

pi2, we prove existence of a weak solutionu.